Download log a (x)

e 的發現始於微分，當
h 逐漸接近零時，計
算 (1+h)1/h之值，其結果無限接近一定值
2.71828...，這個定值就是e ，最早發現此
值的人是瑞士著名數學家歐拉，他以自己姓
名的字頭小寫 e 來命名此無理數。
計算對數函數 y = loga x 的導數，得
dy/dx = (1/x) loga e ，當 a=e 時， loge x 的
導數為1/x，因而有理由使用以 e 為底的對數，
這叫作自然對數。
若將指數函數 ex 作泰勒展開，則得
ex = 1 + x + x2 + x3 + x4 + …
2!
3!
4!
以 x=1 代入上式得
e = 1 + 1 + ½ + 1/6 + 1/12 +….
此級數收斂迅速， e 近似到小數點後 40 位
的數值是
2.71828 18284 59045 23536 02874
The exponential function f with
base a is denoted by f(x)=ax, where
a≠1 , and x is any real number.
The function value will be positive
because a positive base raised to
any power is positive.
Ex: if the base is 2 and x = 4, the
function value f(4) will equal 16. The
graph of f(x)=2x would be (4, 16).
Exponential functions Definition
Take a > 0 and not equal to 1 . Then, the
function defined by
f : R -> R : x -> ax
is called an exponential function with
base a.
Graph and properties
Let f(x) = an exp. fun. with a > 1.
Let g(x) = an exp. Fun. with 0 < a < 1.
From the graphs we see that
The domain is R
 The range is the set of strictly
 positive real numbers
 The function is continuous in its domain
The function is increasing if a > 1 and
decreasing if 0 < a < 1
The x-axis is a horizontal asymptote

Logarithmic functions
Definition and basic properties
Take a > 0 and not equal to 1 . Since the
exponential function f : R -> R : x -> ax are
either increasing or decreasing, the
inverse function is defined. This inverse
function is called the logarithmic function
with base a. We write loga (x)
loga(x) = y <=> ay = x
for x > 0 we have aloga(x) = x
for all x we have loga(ax) = x
Graph
Let f(x) = a logarithmic function with
a > 1.
Let g(x) = a logarithmic function with
0 < a < 1.
log(x.y) = log(x) + log(y)
log(x/y) = log(x) - log(y)
log(xr ) = r.log(x)
Pf:
log(x.y) = u then au = x.y
v
Let log(x) = v then a = x
Let log(y) = w then aw = y
From (1) , (2) and (3)
au = av . aw => au = av + w =>
u=v+w
(1)
(2)
(3)
Change the base of a logarithmic fun.
Theorem:for each strictly positive real
number a and b, different from 1,
loga(x) =(
1
logb(a)
) log.b(x)
Example
Identity
loga(xy) = logax + logay log216 = log28 +
log22
loga(x/y) = logax - logay log2 (5/3) = log25 log23
loga(xr)
logaa
loga1
loga(1/x)
=
r logax
log2(65) = 5 log26
=
=
log22
= 1
log31
= 0
log2(1/3)= -log23
1
0
= -logax
logax = log x = ln x log25 = log 5
log a
ln a
log 2
2.3219
Relationship of the Functions f(x) = logax and g(x) = ax
If a is any positive number, then the
functions f(x) = logax and g(x) = ax are
inverse functions. This means that
alogax= x
for all positive x and
loga(ax) = x
for all real x.
Ex
2log2x=
eln x=
log2(2x) =
ln (ex) =
Ans: x
Definition of Logarithmic Function
For x >0, a>0 , and a ≠ 1, we have
f(x)=loga(x) iff a f(x) =x
Since x > 0, the graph of the above
function will be in quadrants I and
IV.
Comments on Logarithmic Functions
The exponential equation 43=64, could
be written in terms of a logarithmic
equation as log4(64)=3.
The exponential equation 5-2=1/25
can be written as the logarithmic
equation log5(1/25)=-2.
Logarithmic functions are the
inverse of exponential functions.
For example if (4, 16) is a point on
the graph of an exponential
function, then (16, 4) would be the
corresponding point on the graph of
the inverse logarithmic function.
The derivatives of the logarithmic functions
Derivative of logb and ln
d/dx logb(x) = 1 / x ln (b)
An important special case is this:
d/dx ln (x) = 1/x
since ln e =1
Derivative of bx and ex
(d/dx) bx = bx ln(b)
Ex:
d
dx
Ex:
ex - e-x
ex + e-x
d/dx [e
4x2-2
]
Ex: d/dx 2x(4 x ) = 2(4 x ) +2x(4 x ) ln4
Ex: d/dx ln (x2 + 2x -1)
Ex: d/dx ln (3x + 2)
(3x + 2)
Ex: d/dx log
3
(x) = 1 / x ln (3)